U03_L2_T1_we2 Representing Functions as Graphs
U03_L2_T1_we2 Representing Functions as Graphs

Video Transcript

Which of the following graphs represents a function in which the domain is equal to the range?

In this question, we’re given the graphs of five different functions. And we need to determine in which of these functions is the domain equal to the range. To answer this question, let’s start by recalling what we mean by the domain and the range of a function. First, the domain of a function is the set of all input values for our function. Second, we recall the range of a function is the set of all output values for the function given its domain. We need to determine in which of these functions is the domain equal to its range.

And since we’re given the graph of a function, let’s recall how we find the domain and the range of a function from its graph. We do this by remembering that the 𝑥-coordinate of any point on our graph tells us the input value of 𝑥. And the corresponding 𝑦-coordinate gives us the output value for that input value of 𝑥. And therefore, the domain of a function is the set of all 𝑥-coordinates of points on its graph. And the range of a function is the set of all 𝑦-coordinates of points which lie on its graph. So to answer this question, let’s find the domain and range of each of the five options.

Let’s start with option (A), and we’ll start with its domain. First, we can see the lowest value of 𝑥 which is a valid input is when 𝑥 is equal to one because the point with coordinates one, zero is the point with the lowest 𝑥-coordinates which lies on this curve. And for any value greater the one, we can see there’s a point on the curve with this value as an 𝑥-coordinate. For example, the vertical line 𝑥 is equal to five intersects the curve. Therefore, the domain of this function is all values greater than or equal to one.

And We can find the range of this function in the same way. We start by finding the lowest 𝑦-coordinate of a point which lies on the curve. In this case, this is zero. And in fact, we could just stop here because zero is not an element of the domain of our function. So the domain and range are not equal. So the answer is not option (A). However, we can also find an expression for the range of this function. We can see for any value greater than or equal to zero, there is a point on the curve where this is a 𝑦-coordinate. For example, the horizontal line 𝑦 is equal to six intersects our curve. Therefore, the range of this function is all values greater than or equal to zero.

We can follow the same process for all of the other options. In option (B), we can see the lowest value of an 𝑥-coordinate which lies on our curve is zero. And all of the values of 𝑥 greater than or equal to zero lie in the domain of our function. So the domain of this function is all values greater than or equal to zero. We can only input nonnegative values of 𝑥. However, if we find the range of this function, we again have the same problem. Negative one is an output of the function because it’s a 𝑦-coordinate of a point which lies on the curve. And in particular, this function evaluated at zero is equal to negative one. And negative one is not an element of the domain. Therefore, the domain and range of this function are not equal. And if we wanted to, we could just find the range of this function from its diagram. It’s all values greater than or equal to negative one.

We get a very similar story in option (C). Let’s start by finding the domain of this function. First, we notice that is an endpoint of this graph. And we can see that this is the point with the highest 𝑥-coordinate which lies on the graph. This therefore tells us the highest possible input value of the function. The highest input value is zero. And once again, we can see the graph continues infinitely off to the left, which means all values of 𝑥 less than or equal to zero are possible input values of the function. Therefore, the domain of this function is all values less than or equal to zero.

Next, we need to check if the domain and range of this function are equal. And once again, we’re going to use the fact that the range of this function is the set of all 𝑦-coordinates of points which lie on its graph. The lowest 𝑦-coordinate of a point which lies on its graph is the point zero, negative one. The lowest output of this function is negative one. So the domain and range of this function cannot be equal. So option (C) is not correct.

However, we can directly find the range of this function from its diagram. All values greater than or equal to negative one are possible outputs of the function. So the range is the set of values greater than or equal to negative one. We will apply the same process to option (D). We’ll start by finding the domain of this function. We can see the lowest 𝑥-coordinate of a point on the line is the point zero, one with 𝑥-coordinate zero. And since the graph of this function continues infinitely in this direction, we can see that any possible value greater than or equal to zero is a possible input value for our function. The domain of this function is all values greater than or equal to zero.

We can then use this to determine the range of the function. We can see the lowest 𝑦-coordinate of a point on the curve is one which also means the lowest output of the function is one. However, the domain of this function includes zero. Therefore, zero is not an element of the range of this function. The domain and range are not equal. This is enough to exclude this option. However, we can also find the range of this function from the diagram. On the diagram, we can see as 𝑥 approaches infinity, our function also approaches infinity. And therefore, from the diagram, we can see any possible value of 𝑦 greater than or equal to zero is a 𝑦-coordinate on the curve. The range of the function is all values greater than or equal to one.

Finally, let’s determine the domain and the range in option (E). First, the range is the set of all 𝑥-coordinates of points which lie on the curve. We can see the lowest 𝑥-coordinate of a point on the curve is the point zero, zero. And since the graph of our function continues infinitely to the right, there is a point on the curve for every 𝑥-coordinate greater than or equal to zero. Another way of thinking about this is every vertical line in the form 𝑥 is equal to 𝑐, where 𝑐 is greater than or equal to zero, intersects the curve. And the vertical lines to the left of the vertical axis don’t intersect the line. Therefore, the domain is all values greater than or equal to zero.

And we can see something very similar for the range. Remember, the range of the function is the set of all 𝑦-coordinates of points which lie on its curve. And from the diagram, we can see the lowest 𝑦-coordinate of any point on the curve is the point with coordinate zero, zero. And from the diagram, we can see the end behavior of our graph. As 𝑥 is approaching infinity, 𝑦 is unbounded. 𝑦 is approaching positive infinity. So we can see from the diagram any possible value of 𝑦 greater than or equal to zero is the 𝑦-coordinate of a point on the curve. Therefore, the range of this function is all values greater than or equal to zero.

Therefore, we were able to show of the five given options only the graph in option (E) represents a function which has a domain equal to its range.

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